1+1D Quantum Models

TNRKit provides tools to construct partition function tensors for $(1+1)$-dimensional quantum lattice models with translation symmetry (currently only nearest-neighbor Hamiltonians are supported). These tensors can then be studied with any of the 2D tensor network renormalization schemes (TRG, HOTRG, LoopTNR, etc.).

Quantum Partition Functions

A $(1+1)$D quantum system at inverse temperature $\beta$ has partition function

\[\mathcal{Z} = \mathrm{Tr}\, e^{-\beta H}\]

where $H$ is the Hamiltonian. By discretizing the imaginary-time direction into $N$ steps of size $\delta\tau = \beta / N$ and applying a Trotter–Suzuki decomposition, the partition function becomes a contraction of a 2D tensor network. Each time step is represented by a Trotter gate acting on nearest-neighbor pairs.

Tensor Network Representation of $\mathcal{Z}$

When $H = \sum_i h_{i,i+1}$, where $h_{i,i+1} = h$ (due to translation symmetry) is a nearest-neighbor term, the Trotter gate network naturally forms a square lattice. Each Trotter gate $g = \exp(-\delta\tau \, h)$ connects two spatial sites (horizontal direction in the gate diagram) and advances by one imaginary-time step $\delta\tau$ (vertical direction in the gate diagram). However, the gate network is rotated by 45 degrees relative to the standard upright orientation.

gate_to_tensor transforms the rotated gate network into an upright square tensor network. The resulting tensor $T$ is the elementary cell of this upright network, and follows the standard TNRKit leg convention:

The horizontal direction (legs 1 ↔ 4) corresponds to the spatial dimension of the quantum chain, and the vertical direction (legs 2 ↔ 3) corresponds to imaginary time. The imaginary-time step represented by one $T$ tensor is the same $\delta\tau$ as one Trotter gate — the transformation does not coarse-grain in time. In the spatial direction, however, the transformation does renormalize: each $T$ represents two original lattice sites.

Vertical Stacking

To build up the imaginary-time direction, multiple copies of $T$ are stacked vertically. As the stack grows, the bond dimension in the spatial direction would grow exponentially if left unchecked. TNRKit provides two stacking strategies that use HOTRG-style compression to keep the bond dimension under control.

vertical_stack_exp stacks $2^{\text{nfold}}$ copies of $T$ by repeatedly doubling the current stack. Each iteration takes the current stack (representing $2^k$ Trotter steps) and compresses it on top of itself, yielding a stack of $2^{k+1}$ steps. After nfold iterations, the result represents $2^{\text{nfold}}$ Trotter steps. This is expected to be the more accurate approach, since only $m$ iterations are needed to reach $2^m$ steps.
vertical_stack_linear stacks exactly $n$ copies of $T$ linearly (as opposed to a power of 2). Each iteration adds one more copy of the original tensor $T$ on top of the current stack. This is useful when you need a specific number of Trotter steps rather than a power of 2.

Implementing Custom Quantum Models

To construct the partition function of a custom $(1+1)$D quantum Hamiltonian with nearest-neighbor interactions, follow this recipe:

Construct the Trotter gate: Build a 4-leg tensor representing $\exp(-\delta\tau \, h_{i,i+1})$ where $h_{i,i+1}$ is the nearest-neighbor Hamiltonian term. The gate should live in $V \otimes V \leftarrow V \otimes V$, where $V$ is the local Hilbert space.
Convert to partition function tensor: Call gate_to_tensor(gate; trunc=...) to obtain the elementary tensor $T$.
Stack in the time direction: Use vertical_stack_exp or vertical_stack_linear to build up the desired number of Trotter steps.

Now you are ready to feed the resulting 2D tensor network into any of TNRKit's 2D schemes (TRG, HOTRG, LoopTNR, etc.).